8.2. Lowpass filtersBy definition, a lowpass filter is a circuit offering easy passage to lowfrequency signals and difficult passage to highfrequency signals. There are two basic kinds of circuits capable of accomplishing this objective, and many variations of each one: The inductive lowpass filter in Figure below and the capacitive lowpass filter in Figure below
Inductive lowpass filter The inductor's impedance increases with increasing frequency. This high impedance in series tends to block highfrequency signals from getting to the load. This can be demonstrated with a SPICE analysis: (Figure below) inductive lowpass filter v1 1 0 ac 1 sin l1 1 2 3 rload 2 0 1k .ac lin 20 1 200 .plot ac v(2) .end
The response of an inductive lowpass filter falls off with increasing frequency.
Capacitive lowpass filter. The capacitor's impedance decreases with increasing frequency. This low impedance in parallel with the load resistance tends to short out highfrequency signals, dropping most of the voltage across series resistor R_{1}. (Figure below) capacitive lowpass filter v1 1 0 ac 1 sin r1 1 2 500 c1 2 0 7u rload 2 0 1k .ac lin 20 30 150 .plot ac v(2) .end
The response of a capacitive lowpass filter falls off with increasing frequency. The inductive lowpass filter is the pinnacle of simplicity, with only one component comprising the filter. The capacitive version of this filter is not that much more complex, with only a resistor and capacitor needed for operation. However, despite their increased complexity, capacitive filter designs are generally preferred over inductive because capacitors tend to be “purer” reactive components than inductors and therefore are more predictable in their behavior. By “pure” I mean that capacitors exhibit little resistive effects than inductors, making them almost 100% reactive. Inductors, on the other hand, typically exhibit significant dissipative (resistorlike) effects, both in the long lengths of wire used to make them, and in the magnetic losses of the core material. Capacitors also tend to participate less in “coupling” effects with other components (generate and/or receive interference from other components via mutual electric or magnetic fields) than inductors, and are less expensive. However, the inductive lowpass filter is often preferred in ACDC power supplies to filter out the AC “ripple” waveform created when AC is converted (rectified) into DC, passing only the pure DC component. The primary reason for this is the requirement of low filter resistance for the output of such a power supply. A capacitive lowpass filter requires an extra resistance in series with the source, whereas the inductive lowpass filter does not. In the design of a highcurrent circuit like a DC power supply where additional series resistance is undesirable, the inductive lowpass filter is the better design choice. On the other hand, if low weight and compact size are higher priorities than low internal supply resistance in a power supply design, the capacitive lowpass filter might make more sense. All lowpass filters are rated at a certain cutoff frequency. That is, the frequency above which the output voltage falls below 70.7% of the input voltage. This cutoff percentage of 70.7 is not really arbitrary, all though it may seem so at first glance. In a simple capacitive/resistive lowpass filter, it is the frequency at which capacitive reactance in ohms equals resistance in ohms. In a simple capacitive lowpass filter (one resistor, one capacitor), the cutoff frequency is given as:
Inserting the values of R and C from the last SPICE simulation into this formula, we arrive at a cutoff frequency of 45.473 Hz. However, when we look at the plot generated by the SPICE simulation, we see the load voltage well below 70.7% of the source voltage (1 volt) even at a frequency as low as 30 Hz, below the calculated cutoff point. What's wrong? The problem here is that the load resistance of 1 kΩ affects the frequency response of the filter, skewing it down from what the formula told us it would be. Without that load resistance in place, SPICE produces a Bode plot whose numbers make more sense: (Figure below) capacitive lowpass filter v1 1 0 ac 1 sin r1 1 2 500 c1 2 0 7u * note: no load resistor! .ac lin 20 40 50 .plot ac v(2) .end
For the capacitive lowpass filter with R = 500 Ω and C = 7 µF, the Output should be 70.7% at 45.473 Hz. f_{cutoff} = 1/(2πRC) = 1/(2π(500 Ω)(7 µF)) = 45.473 Hz When dealing with filter circuits, it is always important to note that the response of the filter depends on the filter's component values and the impedance of the load. If a cutoff frequency equation fails to give consideration to load impedance, it assumes no load and will fail to give accurate results for a reallife filter conducting power to a load. One frequent application of the capacitive lowpass filter principle is in the design of circuits having components or sections sensitive to electrical “noise.” As mentioned at the beginning of the last chapter, sometimes AC signals can “couple” from one circuit to another via capacitance (C_{stray}) and/or mutual inductance (M_{stray}) between the two sets of conductors. A prime example of this is unwanted AC signals (“noise”) becoming impressed on DC power lines supplying sensitive circuits: (Figure below)
Noise is coupled by stray capacitance and mutual inductance into “clean” DC power. The oscilloscopemeter on the left shows the “clean” power from the DC voltage source. After coupling with the AC noise source via stray mutual inductance and stray capacitance, though, the voltage as measured at the load terminals is now a mix of AC and DC, the AC being unwanted. Normally, one would expect E_{load} to be precisely identical to E_{source}, because the uninterrupted conductors connecting them should make the two sets of points electrically common. However, power conductor impedance allows the two voltages to differ, which means the noise magnitude can vary at different points in the DC system. If we wish to prevent such “noise” from reaching the DC load, all we need to do is connect a lowpass filter near the load to block any coupled signals. In its simplest form, this is nothing more than a capacitor connected directly across the power terminals of the load, the capacitor behaving as a very low impedance to any AC noise, and shorting it out. Such a capacitor is called a decoupling capacitor: (Figure below)
Decoupling capacitor, applied to load, filters noise from DC power supply. A cursory glance at a crowded printedcircuit board (PCB) will typically reveal decoupling capacitors scattered throughout, usually located as close as possible to the sensitive DC loads. Capacitor size is usually 0.1 µF or more, a minimum amount of capacitance needed to produce a low enough impedance to short out any noise. Greater capacitance will do a better job at filtering noise, but size and economics limit decoupling capacitors to meager values.
